arXiv subjects

G. Kaniadakis

Publications and source records attributed to G. Kaniadakis.

At least 19 recordsLinked to original sources

arXiv2024

A graph-space optimal transport FWI approach based on κ-generalized Gaussian distribution

The statistical basis for conventional full-waveform inversion (FWI) approaches is commonly associated with Gaussian statistics. However, errors are rarely Gaussian in non-linear problems like FWI. In this work, we investigate the portability of a new objective function for FWI applications based on the graph-space optimal transport and $κ$-generalized Gaussian probability distribution. In particular, we demonstrate that the proposed objective function is robust in mitigating two critical problems in FWI, which are associated with cycle skipping issues and non-Gaussian errors. The results reveal that our proposal can mitigate the negative influence of cycle-skipping ambiguity and non-Gaussian noises and reduce the computational runtime for computing the transport plan associated with the optimal transport theory.

physics.geo-ph↗

arXiv2024

Relativistic Roots of $κ$-Entropy

The axiomatic structure of the $κ$-statistcal theory is proven. In addition to the first three standard Khinchin--Shannon axioms of continuity, maximality, and expansibility, two further axioms are identified, namely the self-duality axiom and the scaling axiom. It is shown that both the $κ$-entropy and its special limiting case, the classical Boltzmann--Gibbs--Shannon entropy, follow unambiguously from the above new set of five axioms. It has been emphasized that the statistical theory that can be built from $κ$-entropy has a validity that goes beyond physics and can be used to treat physical, natural, or artificial complex systems. The physical origin of the self-duality and scaling axioms has been investigated and traced back to the first principles of relativistic physics, i.e., the Galileo relativity principle and the Einstein principle of the constancy of the speed of light. It has been shown that the $κ$-formalism, which emerges from the $κ$-entropy, can treat both simple (few-body) and complex (statistical) systems in a unified way. Relativistic statistical mechanics based on $κ$-entropy is shown that preserves the main features of classical statistical mechanics (kinetic theory, molecular chaos hypothesis, maximum entropy principle, thermodynamic stability, H-theorem, and Lesche stability). The answers that the $κ$-statistical theory gives to the more-than-a-century-old open problems of relativistic physics, such as how thermodynamic quantities like temperature and entropy vary with the speed of the reference frame, have been emphasized.

cond-mat.stat-mech↗

arXiv2022

Novel predator-prey model admitting exact analytical solution

The Lotka-Volterra predator-prey model still represents the paradigm for the description of the competition in population dynamics. Despite its extreme simplicity, it does not admit an analytical solution, and for this reason, numerical integration methods are usually adopted to apply it to various fields of science. The aim of the present work is to investigate the existence of new predator-prey models sharing the broad features of the standard Lotka-Volterra model and, at the same time, offer the advantage of possessing exact analytical solutions. To this purpose, a general Hamiltonian formalism, which is suitable for treating a large class of predator-prey models in population dynamics within the same framework, has been developed as a first step. The only existing model having the property of admitting a simple exact analytical solution, is identified within the above class of models. The solution of this special predator-prey model is obtained explicitly, in terms of known elementary functions, and its main properties are studied. Finally, the generalization of this model, based on the concept of power-law competition, as well as its extension to the case of $N$-component competition systems, are considered.

q-bio.PE↗

arXiv2022

New power-law tailed distributions emerging in $κ$-statistics

Over the last two decades, it has been argued that the Lorentz transformation mechanism, which imposes the generalization of Newton's classical mechanics into Einstein's special relativity, implies a generalization, or deformation, of the ordinary statistical mechanics. The exponential function, which defines the Boltzmann's factor, emerges properly deformed within this formalism. Starting from this, so-called $κ$-deformed exponential function, we introduce new classes of statistical distributions emerging as the $κ$-deformed version of already known distribution as the Generalized Gamma, Weibull, Logistic which can be adopted in the analysis of statistical data that exhibit power-law tails.

math.ST↗

arXiv2017

Composition law of $κ$-entropy for statistically independent systems

The intriguing and still open question concerning the composition law of $κ$-entropy $S_κ(f)=\frac{1}{2κ}\sum_i (f_i^{1-κ}-f_i^{1+κ})$ with $0<κ<1$ and $\sum_i f_i =1$ is here reconsidered and solved. It is shown that, for a statistical system described by the probability distribution $f=\{ f_{ij}\}$, made up of two statistically independent subsystems, described through the probability distributions $p=\{ p_i\}$ and $q=\{ q_j\}$, respectively, with $f_{ij}=p_iq_j$, the joint entropy $S_κ(p\,q)$ can be obtained starting from the $S_κ(p)$ and $S_κ(q)$ entropies, and additionally from the entropic functionals $S_κ(p/e_κ)$ and $S_κ(q/e_κ)$, $e_κ$ being the $κ$-Napier number. The composition law of the $κ$-entropy is given in closed form, and emerges as a one-parameter generalization of the ordinary additivity law of Boltzmann-Shannon entropy recovered in the $κ\rightarrow 0$ limit.

cond-mat.stat-mech↗

arXiv2016

$κ$-generalized models of income and wealth distributions: A survey

The paper provides a survey of results related to the "$κ$-generalized distribution", a statistical model for the size distribution of income and wealth. Topics include, among others, discussion of basic analytical properties, interrelations with other statistical distributions as well as aspects that are of special interest in the income distribution field, such as the Gini index and the Lorenz curve. An extension of the basic model that is most able to accommodate the special features of wealth data is also reviewed. The survey of empirical applications given in this paper shows the $κ$-generalized models of income and wealth to be in excellent agreement with the observed data in many cases.

q-fin.GN↗

arXiv2013

Theoretical foundations and mathematical formalism of the power-law tailed statistical distributions

We present the main features of the mathematical theory generated by the κ-deformed exponential function exp_κ(x)=(\sqrt{1+κ^2 x^2}+κx)^{1/κ}, with 0<κ<1, developed in the last twelve years, which turns out to be a continuous one parameter deformation of the ordinary mathematics generated by the Euler exponential function. The κ-mathematics has its roots in special relativity and furnishes the theoretical foundations of the κ-statistical mechanics predicting power law tailed statistical distributions which have been observed experimentally in many physical, natural and artificial systems. After introducing the κ-algebra we present the associated κ-differential and κ-integral calculus. Then we obtain the corresponding κ-exponential and κ-logarithm functions and give the κ-version of the main functions of the ordinary mathematics.

math.ST↗

arXiv2012

A generalized statistical model for the size distribution of wealth

In a recent paper in this journal [J. Stat. Mech. (2009) P02037] we proposed a new, physically motivated, distribution function for modeling individual incomes having its roots in the framework of the k-generalized statistical mechanics. The performance of the k-generalized distribution was checked against real data on personal income for the United States in 2003. In this paper we extend our previous model so as to be able to account for the distribution of wealth. Probabilistic functions and inequality measures of this generalized model for wealth distribution are obtained in closed form. In order to check the validity of the proposed model, we analyze the U.S. household wealth distributions from 1984 to 2009 and conclude an excellent agreement with the data that is superior to any other model already known in the literature.

q-fin.GN↗

arXiv2012

Physical origin of the power-law tailed statistical distributions

Starting from the BBGKY hierarchy, describing the kinetics of nonlinear particle system, we obtain the relevant entropy and stationary distribution function. Subsequently, by employing the Lorentz transformations we propose the relativistic generalization of the exponential and logarithmic functions. The related particle distribution and entropy represents the relativistic extension of the classical Maxwell-Boltzmann distribution and of the Boltzmann entropy respectively and define the statistical mechanics presented in [Phys. Rev. E {\bf 66}, 056125 (2002)] and [Phys. Rev. E {\bf 72}, 036108 (2005). The achievements of the present effort, support the idea that the experimentally observed power law tailed statistical distributions in plasma physics, are enforced by the relativistic microscopic particle dynamics.

cond-mat.stat-mech↗

arXiv2011

Power-Law tailed statistical distributions and Lorentz transformations

The present Letter, deals with the statistical theory [Phys. Rev. E {\bf 66}, 056125 (2002) and Phys. Rev E {\bf 72}, 036108 (2005)], which predicts the probability distribution $p(E) \propto \exp_κ (-I)$, where, $I \propto βE -βμ$, is the collision invariant, and $\exp_κ(x)=(\sqrt{1+ κ^2 x^2}+κx)^{1/κ}$, with $κ^2<1$. This, experimentally observed distribution, at low energies behaves as the Maxwell-Boltzmann exponential distribution, while at high energies presents power law tails. Here we show that the function $\exp_κ(x)$ and its inverse $\ln_κ(x)$, can be obtained within the one-particle relativistic dynamics, in a very simple and transparent way, without invoking any extra principle or assumption, starting directly from the Lorentz transformations. The achievements support the idea that the power law tailed distributions are enforced by the Lorentz relativistic microscopic dynamics, like in the case of the exponential distribution which follows from the Newton classical microscopic dynamics.

cond-mat.stat-mech↗

arXiv2010

Relativistic kinetics and power-law tailed distributions

The present paper is devoted to the relativistic statistical theory, introduced in Phys. Rev. E {\bf 66} (2002) 056125 and Phys. Rev. E {\bf 72} (2005) 036108, predicting the particle distribution function $p(E)= \exp_κ (-β[E-μ])$ with $\exp_κ(x)=(\sqrt{1+ κ^2 x^2}+κx)^{1/κ}$, and $κ^2<1$. This, experimentally observed, relativistic distribution, at low energies behaves as the exponential, Maxwell-Boltzmann classical distribution, while at high energies presents power law tails. Here, we obtain the evolution equation, conducting asymptotically to the above distribution, by using a new deductive procedure, starting from the relativistic BBGKY hierarchy and by employing the relativistic molecular chaos hypothesis.

cond-mat.stat-mech↗

arXiv2009

Relativistic Entropy and Related Boltzmann Kinetics

It is well known that the particular form of the two-particle correlation function, in the collisional integral of the classical Boltzmman equation, fix univocally the entropy of the system, which turn out to be the Boltzmann-Gibbs-Shannon entropy. In the ordinary relativistic Boltzmann equation, some standard generalizations, with respect its classical version, imposed by the special relativity, are customarily performed. The only ingredient of the equation, which tacitly remains in its original classical form, is the two-particle correlation function, and this fact imposes that also the relativistic kinetics is governed by the Boltzmann-Gibbs-Shannon entropy. Indeed the ordinary relativistic Boltzmann equation admits as stationary stable distribution, the exponential Juttner distribution. Here, we show that the special relativity laws and the maximum entropy principle, suggest a relativistic generalization also of the two-particle correlation function and then of the entropy. The so obtained, fully relativistic Boltzmann equation, obeys the H-theorem and predicts a stationary stable distribution, presenting power-law tails in the high energy region. The ensued relativistic kinetic theory preserves the main features of the classical kinetics, which recovers in the $c \to \infty$ limit.

physics.class-ph↗

arXiv2009

Maximum entropy principle and power-law tailed distributions

In ordinary statistical mechanics the Boltzmann-Shannon entropy is related to the Maxwell-Bolzmann distribution $p_i$ by means of a twofold link. The first link is differential and is offered by the Jaynes Maximum Entropy Principle. The second link is algebraic and imposes that both the entropy and the distribution must be expressed in terms of the same function in direct and inverse form. Indeed, the Maxwell-Boltzmann distribution $p_i$ is expressed in terms of the exponential function, while the Boltzmann-Shannon entropy is defined as the mean value of $-\ln(p_i)$. In generalized statistical mechanics the second link is customarily relaxed. Here we consider the question if and how is it possible to select generalized statistical theories in which the above mentioned twofold link between entropy and the distribution function continues to hold, such as in the case of ordinary statistical mechanics. Within this scenario, there emerge new couples of direct-inverse functions, i.e. generalized logarithms $Λ(x)$ and generalized exponentials $Λ^{-1}(x)$, defining coherent and self-consistent generalized statistical theories. Interestingly, all these theories preserve the main features of ordinary statistical mechanics, and predict distribution functions presenting power-law tails. Furthermore, the obtained generalized entropies are both thermodynamically and Lesche stable.

cond-mat.stat-mech↗

arXiv2009

A k-generalized statistical mechanics approach to income analysis

This paper proposes a statistical mechanics approach to the analysis of income distribution and inequality. A new distribution function, having its roots in the framework of k-generalized statistics, is derived that is particularly suitable to describe the whole spectrum of incomes, from the low-middle income region up to the high-income Pareto power-law regime. Analytical expressions for the shape, moments and some other basic statistical properties are given. Furthermore, several well-known econometric tools for measuring inequality, which all exist in a closed form, are considered. A method for parameter estimation is also discussed. The model is shown to fit remarkably well the data on personal income for the United States, and the analysis of inequality performed in terms of its parameters reveals very powerful.

physics.soc-ph↗

arXiv2008

The k-generalized distribution: A new descriptive model for the size distribution of incomes

This paper proposes the k-generalized distribution as a model for describing the distribution and dispersion of income within a population. Formulas for the shape, moments and standard tools for inequality measurement - such as the Lorenz curve and the Gini coefficient - are given. A method for parameter estimation is also discussed. The model is shown to fit extremely well the data on personal income distribution in Australia and the United States.

q-fin.GN↗

arXiv2007

k-Generalized Statistics in Personal Income Distribution

Starting from the generalized exponential function $\exp_κ(x)=(\sqrt{1+κ^{2}x^{2}}+κx)^{1/κ}$, with $\exp_{0}(x)=\exp(x)$, proposed in Ref. [G. Kaniadakis, Physica A \textbf{296}, 405 (2001)], the survival function $P_{>}(x)=\exp_κ(-βx^α)$, where $x\in\mathbf{R}^{+}$, $α,β>0$, and $κ\in[0,1)$, is considered in order to analyze the data on personal income distribution for Germany, Italy, and the United Kingdom. The above defined distribution is a continuous one-parameter deformation of the stretched exponential function $P_{>}^{0}(x)=\exp(-βx^α)$\textemdash to which reduces as $κ$ approaches zero\textemdash behaving in very different way in the $x\to0$ and $x\to\infty$ regions. Its bulk is very close to the stretched exponential one, whereas its tail decays following the power-law $P_{>}(x)\sim(2βκ)^{-1/κ}x^{-α/κ}$. This makes the $κ$-generalized function particularly suitable to describe simultaneously the income distribution among both the richest part and the vast majority of the population, generally fitting different curves. An excellent agreement is found between our theoretical model and the observational data on personal income over their entire range.

physics.soc-ph↗

arXiv2006

Towards a relativistic statistical theory

In special relativity the mathematical expressions, defining physical observables as the momentum, the energy etc, emerge as one parameter (light speed) continuous deformations of the corresponding ones of the classical physics. Here, we show that the special relativity imposes a proper one parameter continuous deformation also to the expression of the classical Boltzmann-Gibbs-Shannon entropy. The obtained relativistic entropy permits to construct a coherent and selfconsistent relativistic statistical theory [Phys. Rev. E {\bf 66}, 056125 (2002); Phys. Rev. E {\bf 72}, 036108 (2005)], preserving the main features (maximum entropy principle, thermodynamic stability, Lesche stability, continuity, symmetry, expansivity, decisivity, etc.) of the classical statistical theory, which is recovered in the classical limit. The predicted distribution function is a one-parameter continuous deformation of the classical Maxwell-Boltzmann distribution and has a simple analytic form, showing power law tails in accordance with the experimental evidence.

hep-th↗

arXiv2005

Two generalizations of the Boltzmann equation

We connect two different generalizations of Boltzmann's kinetic theory by requiring the same stationary solution. Non-extensive statistics can be produced by either using corresponding collision rates nonlinear in the one-particle densities or equivalently by using nontrivial energy composition rules in the energy conservation constraint. Direct transformation formulas between key functions of the two approaches are given.

cond-mat.stat-mech↗